Optimal. Leaf size=247 \[ \frac {4 (a x+1)^{5/4}}{a \sqrt [4]{1-a x}}+\frac {5 (1-a x)^{3/4} \sqrt [4]{a x+1}}{a}+\frac {5 \log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{2 \sqrt {2} a}-\frac {5 \log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{2 \sqrt {2} a}-\frac {5 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2} a}+\frac {5 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2} a} \]
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Rubi [A] time = 0.17, antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {6125, 47, 50, 63, 331, 297, 1162, 617, 204, 1165, 628} \[ \frac {4 (a x+1)^{5/4}}{a \sqrt [4]{1-a x}}+\frac {5 (1-a x)^{3/4} \sqrt [4]{a x+1}}{a}+\frac {5 \log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{2 \sqrt {2} a}-\frac {5 \log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{2 \sqrt {2} a}-\frac {5 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2} a}+\frac {5 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2} a} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 204
Rule 297
Rule 331
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 6125
Rubi steps
\begin {align*} \int e^{\frac {5}{2} \tanh ^{-1}(a x)} \, dx &=\int \frac {(1+a x)^{5/4}}{(1-a x)^{5/4}} \, dx\\ &=\frac {4 (1+a x)^{5/4}}{a \sqrt [4]{1-a x}}-5 \int \frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}} \, dx\\ &=\frac {5 (1-a x)^{3/4} \sqrt [4]{1+a x}}{a}+\frac {4 (1+a x)^{5/4}}{a \sqrt [4]{1-a x}}-\frac {5}{2} \int \frac {1}{\sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx\\ &=\frac {5 (1-a x)^{3/4} \sqrt [4]{1+a x}}{a}+\frac {4 (1+a x)^{5/4}}{a \sqrt [4]{1-a x}}+\frac {10 \operatorname {Subst}\left (\int \frac {x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-a x}\right )}{a}\\ &=\frac {5 (1-a x)^{3/4} \sqrt [4]{1+a x}}{a}+\frac {4 (1+a x)^{5/4}}{a \sqrt [4]{1-a x}}+\frac {10 \operatorname {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{a}\\ &=\frac {5 (1-a x)^{3/4} \sqrt [4]{1+a x}}{a}+\frac {4 (1+a x)^{5/4}}{a \sqrt [4]{1-a x}}-\frac {5 \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{a}+\frac {5 \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{a}\\ &=\frac {5 (1-a x)^{3/4} \sqrt [4]{1+a x}}{a}+\frac {4 (1+a x)^{5/4}}{a \sqrt [4]{1-a x}}+\frac {5 \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{2 a}+\frac {5 \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{2 a}+\frac {5 \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{2 \sqrt {2} a}+\frac {5 \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{2 \sqrt {2} a}\\ &=\frac {5 (1-a x)^{3/4} \sqrt [4]{1+a x}}{a}+\frac {4 (1+a x)^{5/4}}{a \sqrt [4]{1-a x}}+\frac {5 \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{2 \sqrt {2} a}-\frac {5 \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{2 \sqrt {2} a}+\frac {5 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt {2} a}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt {2} a}\\ &=\frac {5 (1-a x)^{3/4} \sqrt [4]{1+a x}}{a}+\frac {4 (1+a x)^{5/4}}{a \sqrt [4]{1-a x}}-\frac {5 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt {2} a}+\frac {5 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt {2} a}+\frac {5 \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{2 \sqrt {2} a}-\frac {5 \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{2 \sqrt {2} a}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 174, normalized size = 0.70 \[ \frac {\frac {40 e^{\frac {1}{2} \tanh ^{-1}(a x)}}{e^{2 \tanh ^{-1}(a x)}+1}+\frac {32 e^{\frac {5}{2} \tanh ^{-1}(a x)}}{e^{2 \tanh ^{-1}(a x)}+1}+5 \sqrt {2} \log \left (-\sqrt {2} e^{\frac {1}{2} \tanh ^{-1}(a x)}+e^{\tanh ^{-1}(a x)}+1\right )-5 \sqrt {2} \log \left (\sqrt {2} e^{\frac {1}{2} \tanh ^{-1}(a x)}+e^{\tanh ^{-1}(a x)}+1\right )+10 \sqrt {2} \tan ^{-1}\left (1-\sqrt {2} e^{\frac {1}{2} \tanh ^{-1}(a x)}\right )-10 \sqrt {2} \tan ^{-1}\left (\sqrt {2} e^{\frac {1}{2} \tanh ^{-1}(a x)}+1\right )}{4 a} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.85, size = 512, normalized size = 2.07 \[ \frac {20 \, \sqrt {2} a \frac {1}{a^{4}}^{\frac {1}{4}} \arctan \left (\sqrt {2} a^{3} \sqrt {\frac {\sqrt {2} {\left (a^{2} x - a\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {1}{4}} + {\left (a^{3} x - a^{2}\right )} \sqrt {\frac {1}{a^{4}}} - \sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {3}{4}} - \sqrt {2} a^{3} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {3}{4}} - 1\right ) + 20 \, \sqrt {2} a \frac {1}{a^{4}}^{\frac {1}{4}} \arctan \left (\sqrt {2} a^{3} \sqrt {-\frac {\sqrt {2} {\left (a^{2} x - a\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {1}{4}} - {\left (a^{3} x - a^{2}\right )} \sqrt {\frac {1}{a^{4}}} + \sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {3}{4}} - \sqrt {2} a^{3} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {3}{4}} + 1\right ) - 5 \, \sqrt {2} a \frac {1}{a^{4}}^{\frac {1}{4}} \log \left (\frac {\sqrt {2} {\left (a^{2} x - a\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {1}{4}} + {\left (a^{3} x - a^{2}\right )} \sqrt {\frac {1}{a^{4}}} - \sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right ) + 5 \, \sqrt {2} a \frac {1}{a^{4}}^{\frac {1}{4}} \log \left (-\frac {\sqrt {2} {\left (a^{2} x - a\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {1}{4}} - {\left (a^{3} x - a^{2}\right )} \sqrt {\frac {1}{a^{4}}} + \sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right ) - 4 \, {\left (a x - 9\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{4 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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